Floyd's Algorithm for Finding Shortest Paths in a Graph, Slides of Design and Analysis of Algorithms

Download Floyd's Algorithm for Finding Shortest Paths in a Graph and more Slides Design and Analysis of Algorithms in PDF only on Docsity!

Floyd’s Algorithm

All pairs shortest path

Floyd’s Algorithm 2

All pairs shortest path

• The problem: find the shortest path between every pair of vertices of a graph
• The graph : may contain negative edges but no negative cycles
• A representation : a weight matrix where W(i,j)=0 if i=j. W(i,j)= if there is no edge between i and j. W(i,j)=“weight of edge”

Floyd’s Algorithm 4

The subproblems

• How can we define the shortest distance di,j in terms of “smaller” problems?
• One way is to restrict the paths to only include vertices from a restricted subset.
• Initially, the subset is empty.
• Then, it is incrementally increased until it includes all the vertices.

Floyd’s Algorithm 5

The subproblems

• Let D ( k )[ i,j ]=weight of a shortest path from vi to vj using only vertices from { v 1 , v 2 ,…, vk } as intermediate vertices in the path

– D (0)= W

• D ( n )= D which is the goal matrix
• How do we compute D ( k )^ from D ( k -1)^?

Floyd’s Algorithm 7

Example

W = D^0 =

P =

Floyd’s Algorithm 8

D^1 =

P =

5

(^42)

k = 1

Vertex 1 can

be intermediate

node

D^1 [2,3] = min( D^0 [2,3], D^0 [2,1]+D^0 [1,3] ) = min (, 7) = 7

D^1 [3,2] = min( D^0 [3,2], D^0 [3,1]+D^0 [1,2] ) = min (-3,) = -

D^0 =

Floyd’s Algorithm 10

D^3 =

P =

D^3 [1,2] = min(D^2 [1,2], D^2 [1,3]+D^2 [3,2] ) = min (4, 5+(-3)) = 2

D^3 [2,1] = min(D^2 [2,1], D^2 [2,3]+D^2 [3,1] ) = min (2, 7+ (-1)) = 2

D^2 =

5

(^4 )

k = 3

Vertices 1, 2, 3

can be

intermediate

Floyd’s Algorithm 11

Floyd's Algorithm: Using 2 D matrices

Floyd

1. D  W // initialize D array to W [ ]

2. P  0 // initialize P array to [0]

3. for k  1 to n // Computing D’ from D

4. do for i  1 to n

5. do for j  1 to n

6. if ( D [ i , j ] > D [ i , k ] + D [ k , j ] )

7. then D’ [ i , j ]  D [ i , k ] + D [ k , j ]

8. P [ i, j ]  k ;

9. else D’ [ i , j ]  D [ i , j ]

10. Move D’ to D.

Floyd’s Algorithm 13

The k th column

• k th column of Dk^ is equal to the k th column of Dk -
• Intuitively true - a path from i to k will not become shorter by adding k to the allowed subset of intermediate vertices
• For all i, D(k)[i,k] = = min{ D(k-1)[i,k], D(k-1)[i,k]+ D(k-1)[k,k] } = min { D(k-1)[i,k], D(k-1)[i,k]+0 } = D(k-1)[i,k]

Floyd’s Algorithm 14

The k th row

• k th row of Dk^ is equal to the k th row of Dk -

For all j , D ( k )[ k,j ] = = min{ D ( k -1)[ k,j ], D( k -1)[ k,k ]+ D ( k -1)[ k,j ] } = min{ D ( k -1)[ k,j ], 0+ D ( k -1)[ k,j ] } = D ( k -1)[ k,j ]

Floyd’s Algorithm 16

Example